Upvote:6
From AirBnB Q&A:
If a host asks you for more money than what you paid on the site and the extra charge wasn’t stated in the listing or in the message thread, you can dispute the charges in the Resolution Center. Never pay a host directly for these charges.
and also
Can a security deposit be collected outside of the Airbnb website?
No. To make sure guest payments are protected and to ensure that hosts are properly compensated for reservation damages, we ask that all security deposits be handled through the Airbnb website. Payments outside of the Airbnb website, including security deposits, are a violation of our Terms of Service.
The host can make a claim ony your security deposit via airbnb, he doesn't need your card.